Saturday, January 11, 2014

Alexandrov topology

Given a preorder we can form a special kind of topology called an Alexandrov topology and given such a topology we can go back to the preorder so we have a bijection between the class of preorders and the class of Alexandrov topologies. Here is an example of Alexandrov topology formed by a partial order:
(= (alexandrov-topology (weak-order [#{0} #{1} #{2}]))
     #{0 1 2}
     #{#{} #{2} #{1 2} #{0 1 2}}))
Using the specialization preorder function we can specify classes of Alexandrov topologies that correspond to classes of preorders in our ontology. Here are a few such classes of topologies:
(def partition-topology?
  (comp equivalence-relation? specialization-preorder))

(def discrete-topology?
  (comp antichain? specialization-preorder))

(def trivial-topology?
  (comp complete-relation? specialization-preorder))
Topology is based upon the inclusion order of sets provided by the ontology and the specialization preorder so I think that it is fair to say that topology is based upon order theory.

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